[MUSIC] Hello everyone and welcome to field exercise number two focus group. I am sure, you've already discovered that field exercise two, looking for playing groups is much tougher than looking for the point groups. And we should begin by reminding ourselves what was needed for this field exercise. First of all, you have to identify three objects in three different plane groups; you're not allowed to get two objects in the same plane group, you'll, otherwise you'll lose marks. Secondly, you need to identify all of the symmetry operators for that plane group. Then you have to label them correctly on your photograph. Now, this is a very important field exercise, especially for the track one students because this exercise is worth 30% of the overall grading, so you really have to do take this one seriously. Now, what I'd like to do it give you some hints and tips before we begin, so that you know how to approach this exercise. The first and most important thing is that you download this file and this is a list of all 17 playing groups and I would strongly advise that everybody prints it out and has it next to them when they try to analyze their photographs. The second thing to notice is that in plane symmetry, you will either have rotation points or you will have mirror lines, or you will have glide lines. Now one of the tricks to note is that the rotation points, if they're present, will always be at the origin of the unit cell. You will never find them scattered through the unit cell. They will either be at the origin or if it's a centered cell, they will be at the center, but never anywhere else. So if you have an image or an object showing rotation, that must be the origin. If you have two types of rotation, say twofold and fourfold together, then the highest order rotation is what is found at the origin. So it'd be fourfold at the origin and not twofold or if you had another object which had six fold and three fold rotation, the six fold must be at the origin, so that can be your starting point for looking for the unit sills. So let's begin by asking the group a people to talk about their photographs and we'll start with Hui Min. >> Okay, so basically for my part. I found a pattern in the NTUC Supermarket and it's found on the bathroom mat. So for this pattern, I've chosen a plane symmetry group, which is p 6 mm and it is the number 17 of them. So, for p 6 m m, there is the mirror line, the glide lines, twofold rotation, threefold rotation and the sixfold rotation. >> I've been working on WeiMing's work, I, You're choosing the bathroom tiles on, did you find in NTUC Supermarket and I can see that the symmetry operations that you put on the picture is correct. So, I'm giving you two marks for the positioning correctly. I can see that you identified all the symmetry in the picture you put, so I will give you one mark. And then I also see that you put all the symmetry operations correctly, so I will be giving you two marks. So, in total, you have five marks. Congratulations. >> All right, so thank you very much, Martin, for your comments on Hui Min's photograph. This is actually quite a tricky patent. And also we have to highlight the difference between the unit's cell which might describe the entire tessellation or patent and also the internal symmetry that, that describes the patent. In this case, if we look at the unit cell which is hexagonal then we can use it to describe the entire tessellation. If I was to copy the unit cell, say four copies, then clearly, I can stack those unit cells together, so that they are touching on all their edges. There is no spaces between. And you can see that every origin of that unit cell is the same, and so it replicates the entire pattern. But the other characteristic to choose the hexagon unit cell is that you must be able to take points on the pattern and right type them in increments of 60 degrees and leave the pattern unchanged. And in this case we're unable to do that. So the unit cell is good, but it doesn't reflect the internal symmetry of the pattern. So how do we get around that? Well, I think the other way to look at this pattern is to consider the large open circles. And if we do that, you might think that you can pick out a square pattern. So, let's begin with that, [SOUND]. We could, for example, choose a square here. And equally, if we would to take squares and stack them together, we'll create a replicate of the entire pattern. Now once we choose a square, then intuitively you might think it has be a p 4 type pattern, that would be the logical thing to do. And that implies that when you rotate the pattern through increments of 90 degrees, it will be unchanged. So if we were to take this pattern and copy it [SOUND] And rotate it through 90 degrees, does the pattern look the same? Actually it doesn't, what you can see now is that the way the lines that were horizontal have be, moved to be vertical. So it can't p 4, even though the unit cell looks to be square. So if it's not p 4, we have to move to a lower symmetry. So that implies having two fold rotation and if we were to rot, to rotate through 180 degrees, then you can see the pattern remains unchanged. So we know that there must be two fold rotation here. So let's work on the enlarged pattern. We'll take away the hexagonal cell and now let's insert the square unit cell on top. Now clearly if you look at the origin of this unit cell, you have a two fold rotation point. So, we can begin by inserting those. If you look at the plane group diagrams, you should also have twofold rotation at the center of the unit cell as well. So if we examine p 2 m m, or p 2 m g, p 2 g g or c 2 mm, all of the have twofold rotations in the center. So we can put twofold rotation there and double check that it's operating. Now then, the next easiest thing to look for are the mirror lines. And I think you'd all agree that it's quite obviously a mirror line horizontal across the center. There's also a mirror line vertically across the center. So, if it's p 2 m m, that's the complete description of the unit cell, but let's check for some of the unit cells with higher symmetry. If we look at c 2 mm, then in this case it does have the vertical and horizontal mirrors. So now we're left with 2 choices, p 2 m m, or c 2 mm. Now, when we look at c 2 mm, we can see that in the top quadrant there's another twofold rotation points. So we have to check, is that there. So let's put another twofold rotation, in here. If we rotate around this point do we leave the pattern unchanged, when we rotate by 180 degrees? In fact we do and that will be found in all of the quadrants of the unit cell. So, we can put those in. Now, once we've got to that point, we know that the answer has to be c 2 mm, we've excluded p 2 mm because it doesn't have those additional twofold rotation points. Then we finally check for the c 2 m symmetry diagram. Do we have the additional glide lines? The glide lines intersect at these two forward rotation points. So lets get some glide lines in there. Maybe we look at the horizontal glide first. Remember with a glide you reflect across a glide line and you'll translate through half the size of the unit cell half the length of the unit cell. So for example if we took this little square here. We want to reflect across and translate by half the unit cell. And then we end up with this square here. So we go from the square shown here, reflect, translate, reflect, translate and we get back to where we started. So the horizontal glides are working. If we have a look at the vertical glides, same thing. So, the correct answer for this pattern, is c 2 mm, and it highlights very nicely, this interaction between shape of the unit cell, and the internal symmetry. In this case, the, the pattern is particularly interesting because if you look at it quickly, maybe it's hexagonal because you can stack up the hexagonal cells as what she did. But of course, you can't get the symmetry. You look at another level you think maybe it's a square pattern because you have the large open circles. And you might think it's got to be a fourfold rotation operating. But if you look in the detail of the pattern, it can be fourfold, it has to be twofold. So this kind of, this cascading logic when you solve these problems to determine a symmetry. You find the unit cell and you see if the symmetry of the pattern matches it and then you keep working forward until you've got the major parts of the symmetry sorted out and then you work towards the highest symmetry possible. So in our case, we started looking at p 2 mm which is number 6 and then there are three other possibilities, seven, eight and nine. And we keep cascading up, until we get the highest symmetry which fits this pattern, in this case it's c 2 mm, which is number nine. So, a very interesting example. In terms of the marking, for this, I would say that you did choose a unit cell which could replicate the entire pattern. So, I would give you the full two marks for that. You identified all of the symmetry operators correctly for p6mm. So, another mark for that. But because that symmetry did not overlay correctly on a pattern, I give you zero for that. So I would say the correct marking for this would be three out of five. All right, so let's now move on to the next example of these photographs and the symmetry interpretation. >> Okay, so for my part I actually I took the photos of tiles used for paving at Takashimaya Department Store. And for the tiles that I took, the unit cell shows two mirror lines. One at a vertical and the other one at the horizontal. As well as the glide lines, twofold rotation and fourfold rotation. So for that, I grouped the unit cell as p 4 m m, which is number 11 of the plane symmetry group. >> So I feel that Wanzhen has a good and classic example of tiling. And I felt that she identified p 4 mm as the correct plane symmetry group. So for identifying the correct position of the unit cell, I'll give her two marks for that and for identifying the symmetry operators with mirror lines, glide lines, 2 fold rotation and 4 fold rotation, she would get one mark for that. And for super imposing of this symmetry operators on the diagram appropriately, I'd have to give her two marks for that and she would get a total of five marks. >> Thank you very much Zi Ying. I think that's a perfect analysis on marking for Wangzhen's, photograph, so congratulations, I would agree with five out of five for that effort. Well done. So now we should see what Wilson has come up with for his photograph in his symmetry analysis. >> The symmetry that I've chosen to show the students watching the video is a brick wall. And this brick wall can be see throughout the wall. And the location in which I have taken this brick wall is at Yew Tee MRT station. And for my symmetry operators is a two fold rotation and my plane groups symbol is, p 2. >> Okay, for Wilson, your unit cell does, that's it correctly and there are the symmetry operators there as well, so, can give you the marks for that. Two marks for the position unit cell, and one mark for all the symmetry operators. However, the pattern does not really give the overall symmetry of the pattern. What you have drawn is a parallelogram. P 2, however if you had drawn something like this, that would actually give you a p 2 mm instead of a p 2. So and as this has more symmetry than a p 2, this would be the actual plane group symbol. So I cannot give you any marks for that. So, the total would be three out of five for this plane symmetry. >> Thank you very much Yi Xun, for that analysis of Wilson's photograph. Let's work through the fundamentals for this analysis. If we look at the unit cell that was selected. Now, if we have a look at this particular unit cell that Wilson selected. Let's just put a circle around each of the corners or one of the corners. What you can see is that pointing down from a corner, you have the joint in a brick. If we go down to this corner here, you see the joint points up. So, in fact every corner is not the same for the unit cell selected. So it cannot be the unit cell, it is not a repeating unit. If you wanted to try and use a shade that looked a little like that, you would have to extend the unit cell down to here, or where should I go, perhaps to, yeah, you would have to. Can you do it at all? Yes, you would have to extend the unit cell to here and here. So now it points down everywhere. So now every corner of that unit cell, you will see the joint points down, so that could be a valid unit cell. But then we come to the issue of has the internal symmetry being shown correctly, and both Wilson and Yi Xun found twofold rotation points, which is correct. So, this unit cell here actually implies that perhaps it's a p 1, number one, plain group, with no symmetry inside. But, we already know there's some twofold rotation, so this can't be right either. Now, what Yi Xun came up with is making a rectangular unit cell. So let's try that. [SOUND] And let's perhaps start here at the bottom of a joint. Go to the next bottom of the joint in the bricks go across to here and that might be a good unit cell. [SOUND] So now, you can see every origin looks the same. But the other thing to consider now, is having established that the unit cell, where do we position it on top of the pattern, so that it will be consistent with one of the 17 plane groups? Remember that for the plane groups formally you will have rotational points at the origin. So we know this twofold rotation. So for example, if we position the unit cell here, then what you have is twofold rotation at every point. So let me show you this on the enlarged pattern so it becomes clearer. We can put twofold rotation point here. So far, so good. Now we begin to look for the mirror lines. We can see that there is going to be both a horizontal and a vertical mirror. And we can see that when we have a vertical and a horizontal mirror it can be consistent with p 2 m m, not good for p 2 m g, not good for p 2 g g, consistent with c 2 mm. So we're back to this selection p 2 mm or c 2 mm, is it centered or primitive? Now, the way to decide is do we have a two fold rotation at the center of each quadrant of the unit cell? That's the key difference between the primitive and the centered cell. In fact, if you inspect, you can see that we do have two fold rotation. And then by inspection of the symmetry diagram for c 2 mm, you can see there must be additional horizontal and vertical glide lines and they could be put in at the end. So that would be a correct answer for this particular brick, brick work pattern. So what we should actually put up here, [SOUND] is c 2 mm, remembering that for applying group the leading letter is always lowercase. And it will have to be plane group number 9, in this case the unit cell selected actually can't replicate the entire pattern because remember the joint pointed up and down. So, I think maybe I'd give a mark, but it's not really right, it's it may be a mark for that. You weren't able to identify all of the symmetry operators on top of that unit cell, if we look at the, you'd suggested p 2, if we have a look at, p 2, they should also be twofold rotation in the edges the unit cell that you didn't show. So I'd probably give a zero for that, and then it's clearly not correct. So, in this case I would have given you one out of five for this analysis of the pattern. but, again, don't feel frustrated this really takes quite a lot of thought to solve these plane group patterns. Again, you'll always trying to move towards the highest symmetry, that's satisfies that particular pattern. Okay, so now we've looked at what group A have done. It's group B's turn, so we'll find out what sort of pattern you came up with. >> I took a picture of one museum in South Korea,. I realize that the museum has a wall frame that represents plane symmetry there. I analyzed the, that the wall frame has a symmetry operations of mirror lines, and twofold rotations. They have two types of mirror lines, horizontal mirror lines and vertical mirror lines so the plane symmetry group should be in b 2 m m. >> Okay with Martin Mardjuki's pattern. So I'm very impressed with the pattern because it looks very complicated. But yeah, he can find, very simple symmetry, group to it. So, I'll give you two marks for drawing of the position of the units out correctly because, it is correct and I'll give you one mark for identifying all the symmetry operators, as seen on the right. And also, you will get two marks for superimposing a symmetry diagram with all symmetry operators correctly, so you have got a total of five out of five, which is the full marks. >> Okay, thank you very much for your really complicated pattern Martin and also the analysis from Hui Min, but I think if we look more carefully, we can again find more twofold rotation points. And then, that basically locks in, that we must be using the c 2 mm point group, so that also implies there have to be vertical and horizontal glide lines. We then put in the vertical glides. So, the correct answer in this case, will be c 2 mm, so. For the marking here, drawing the position of the unit cell correctly, two out of two, I would agree. Identifying all the symmetry operators for p 2 mm, you did identify them correctly, so I'd give one for that, but you didn't quite get the right space group, you were basically one step away. You did the primitive, rather than the centered. So, I'd give you one out of two. For that, so 4 out of 5 for the total marking this case. But a very complicated pattern, so well done. To, to identify the symmetry element as well as you did. So let's move on to the next example from group B. >> For my example on the plane symmetry group, I've identify the color tiles in my home. And I use the plane symmetry group of p 4 mm, which is number 11. From the identified unit cell, I could impose a symmetry operator such as mirror lines, glide lines, twofold rotation, and fourfold rotation in the symmetry operation diagram as shown. >> Thank you Zi Ying for sharing your beautiful tiling at home, while for your example, you have correctly drawn the position of the unit cell, so two marks given for that, also you have correctly identified the symmetry operators so we mark you from that. And lastly, you have correctly superimposed the symmetry diagram with all the symmetry operators, so two marks for that. You'll get a total of five out of five. >> Thank you very much Zi Ying, that really was a beautiful example and I think it was very nicely marked as well. Five out of five is certainly deserved there. I really have no other comments. So let's go on to our final photograph, and symmetry analysis. >> Okay, for my photograph, I took a picture of the glass walkway outside B2 of MSE building. This photograph has very, a lot of small, little circles around it. And it forms a very nice plain symmetry. This plain symmetry should belong in group p 2. Thank you Yi Xun for that wonderful example, you have taken photo of. It may seem like a very easy symmetry shape, but it can be quite tricky. So you have chosen p 2 to be, to be your unit cell, unfortunately if you were to draw the circle inside the unit cell, you will have an infinite fold rotation, not just two fold rotation. So as far as drawing the position of unit cell correctly, I can not give you that two marks. For identifying all the symmetry operators given for your p 2, you have to identified all your symmetry operators an I will give you one mark for that. However, super imposing on the symmetry diagram for all symmetry operators correctly, I cannot give you full marks for that. So, because, no matter how you rotate it, you will not get the, a p 2. To greet you, to much for drawing the, the position unit cell correctly I couldn't give you because you've chosen a p 2, but unfortunately it's a infinite full rotation. I'll give you one mark for identifying all the symmetry operators. But I also can't give you another two more marks for superimposing the symmetry diagram with all the symmetry operators correctly, so in total you only have one out of five marks. >> All right, so thank you very much, Yi Xun, for that example, and also Wilson's marking. I would agree that the unit cell, that Yi Xun provided is correct. We can replicate the whole pattern because it's got one circle in the center. So that's clearly correct. But actually there's more and and higher fold rotation in this pattern, than is indicated. If we jump directly, to the enlarged part of the pattern, so let me just take away what we have there at the moment. And lets enlarge this a little bit more. And you can see if you look at the origin, you have a surrounding of one, two, three, four bright areas. So what we actually have at the corner here, is fourfold rotation. We can also see straight away that there are mirror lines and there are horizontal and vertical mirrors, so we can now inspect the diagrams given playing group tables. When we do that, we can see that p 4 has to be excluded, no mirror lines, so that's not right. If we look at p 4 mm, yes, there are mirrors going through the center horizontally and vertically, so that's a possibility. And then we have p4 gm but no mirrors passing through the center. So, we now know that the correct answer must be p4 mm and we can check if the other symmetry is operating. For p 4 mm, there have to be twofold rotation points in the edges of the unit cell. In addition for p 4 mm, there are also, glide lines, diagonally in the unit cell and we can introduce those. [SOUND] So, in this case, what we can say is that the correct plane group, would be p 4 mm, which is number 11. So to mark this I would actually give 4 marks for identifying the unit cell, because that can replicate the entire pattern. Then can it comes to identifying the symmetry operators and you chose p 2, it wasn't the correct one. But you put the overlay of the symmetry operators correctly, so I'd also give a mark for that. In terms of superimposing the symmetry elements, I would actually give you one mark for that because you identified the twofold rotation, you failed to identify the four one, the fourfold rotation. So, I think you got part of the way to that answer, so in this case I would have given you four out of five for this effort. The key points that came out of this exercise, is that you must refer back to the 17 plane groups when you do this exercise. You have to end up with an answer consistent with one of those playing groups, that's the first point. The second thing is always look to begin with for rotation points. These tend to be most obvious. And once you find a rotation point, that must be at the origin of the unit cell. If you have two types of rotation operating, say twofold and fourfold, the higher fold rotation, fourfold in this case, would be at the origin of the unit cell. And finally you should check that all of the symmetry which is shown on the symmetry diagram is in fact operating on your pattern. So thank you very much for all your help. And I hope the students who are watching this can now complete their homework, much more quickly and confidently. Thank you again.
